Example
A simple example of (within-day) DTA is as follows. Figure 8.1 depicts a road network with two nodes and two links. Node A represents the origin and node B represents the destination. The links represent the highways connecting the origin and the destination. Any driver can go from A to B by car via one of the two routes, that is via either Link 1 or Link 2. However, the minimum travel time via Link 2 is 30 minutes less than that via Link 1. If the arrival rate of vehicles at the bottleneck in the middle of Link 2 at any instant is not greater than the capacity (i.e. the maximum number of vehicles that can pass through the bottleneck per hour) of 2000 vehicles per hour, all vehicles can pass through the bottleneck without delay and their travel time is 30 minutes. Otherwise, a queue is formed behind the bottleneck and the travel time via Link 2 is increased. The longer the queue, the higher the travel time. There is also a bottleneck in Link 1 with a capacity of 4000 vehicles per hour and the minimum travel time via Link 1 is 1 hour. It is known that between 6:00 am and 10:00 am, a total of 8000 drivers travel to B from A along both routes. All of these drivers must reach B on or before 9:00 am. These drivers have a choice of departure time in addition to link (or route). They can select a departure time so that they arrive at B at 9:00 pm sharp but they waste a lot of time in queuing. They can also depart early to have less queuing time (or waiting time in queue) and arrive at B early. However, arriving at B too early is not desirable as the time between the arrival time and 9:00 am is wasted at B, because the time can be reserved for other
Figure 8.1: Example network.
activities. Given that the demand from A to B during that period is 8000 vehicles, the problem is to determine the numbers of vehicles using Links 1 and 2 over the study period. In other words, the problem is to find out the time-varying demand splits. Note that the splits depend on the travel times on both links and the travel times also depend on the splits.
Figure 8.2 represents a solution for this simple example. The cumulative arrival curves represent the total number of drivers entering the links over time, whereas the cumulative departure curves represent the total number of drivers leaving the links over time. The vertical distance between the cumulative arrival and departure curves at a particular time gives the number of vehicles on the link at that time. The horizontal distance between two curves gives the travel time of a particular driver. As the minimum travel time on Link 2 is initially less than that on Link 1, drivers initially select Link 2. As the arrival rate of Link 2 is greater than the capacity of the bottleneck, the travel time on this link increases until it is equal to the minimum travel time of Link 1 of 60 minutes. Then, both links are chosen by drivers and all drivers can reach B before 9:00 am. As can be seen in Figure 8.2, the vertical distance between the cumulative arrival and departure curves is changing over time, meaning that the numbers of vehicles on the two links are changing over time. This is because the queuing time changes over time, which affects the departure time and route choice of drivers. The end result is that the minimum travel time from A to B is 30 minutes and the maximum travel time is 90 minutes. Some drivers depart earlier to have less travel time and queuing time. The first driver to leave A can travel to B without facing congestion and arrives at B at 7:00 am, whereas the last driver leaves A at 7:30 am and requires a travel time of 90 minutes to arrive at B sharply at 9:00 am. Furthermore, the number of drivers using each of the two links (which is the height of the curve) is equal to 4000 and the sum is equal to the demand of 8000 vehicles. However, in general, the usage of each link may not be the same.
This example also illustrates that DTA consists of two main components, namely travel choice and traffic flow. The travel choice component determines the traffic flow level on each road at each instant of time, given the road network performance in terms of the time-varying travel times on each road. The traffic flow component depicts how vehicular traffic propagates inside a road network, given the demand split to each route over time, and governs the performance of the road network, in the sense that more traffic on a link results in a higher travel time. The output of
Figure 8.2: Cumulative arrivals and departures on Links 1 and 2 (Szeto & Wong, 2011).
the travel choice component is the input of the traffic flow component while the output of the traffic flow component is the input of the travel choice component. DTA is then used to determine the flow pattern that satisfies the two components simultaneously.
In this example, the demand of 8000 vehicles is required to propagate on either one of the two links. The travel choice component determines the split of the demand (or the number of vehicles entering each link) over time, based on the time- varying travel time of each link. Then, the traffic flow component propagates each vehicle on the respective links and determines the travel time. The travel time on each link over time must be the same as that used to determine the time-varying demand splits if the splits are optimal. Otherwise, a new set of splits based on the output of the traffic flow component should be used to determine the traffic flow on each link over time.
